[Pkg-privacy-commits] [golang-ed25519-dev] 02/11: Replace implementation with a port of ref10 code.
Ximin Luo
infinity0 at moszumanska.debian.org
Sat Aug 22 12:55:25 UTC 2015
This is an automated email from the git hooks/post-receive script.
infinity0 pushed a commit to branch master
in repository golang-ed25519-dev.
commit 5613cf01b3df739fcda8f197edaf754fa9e627b1
Author: Adam Langley <agl at imperialviolet.org>
Date: Mon Sep 2 11:53:39 2013 -0400
Replace implementation with a port of ref10 code.
This implementation is much faster and is now constant-time.
---
const.go | 1403 ++++++++++++++++++++++++++++++++++
ed25519.go | 2242 +++++++++++++++++++++++++++++++++++++++++++++++++++++++
ed25519_ref.go | 377 ----------
ed25519_test.go | 3 +-
4 files changed, 3646 insertions(+), 379 deletions(-)
diff --git a/const.go b/const.go
new file mode 100644
index 0000000..e534b55
--- /dev/null
+++ b/const.go
@@ -0,0 +1,1403 @@
+package ed25519
+
+var d = fieldElement{
+ -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116,
+}
+
+var d2 = fieldElement{
+ -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199,
+}
+
+var sqrtm1 = fieldElement{
+ -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482,
+}
+
+var bi = [8]preComputedGroupElement{
+ {
+ fieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605},
+ fieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378},
+ fieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546},
+ },
+ {
+ fieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024},
+ fieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574},
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+ },
+ {
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+ fieldElement{-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222, 9677543, -32294889, -6456008},
+ },
+ {
+ fieldElement{-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579, -7839692, -7852844, -8138429},
+ fieldElement{-15236356, -15433509, 7766470, 746860, 26346930, -10221762, -27333451, 10754588, -9431476, 5203576},
+ fieldElement{31834314, 14135496, -770007, 5159118, 20917671, -16768096, -7467973, -7337524, 31809243, 7347066},
+ },
+ {
+ fieldElement{-9606723, -11874240, 20414459, 13033986, 13716524, -11691881, 19797970, -12211255, 15192876, -2087490},
+ fieldElement{-12663563, -2181719, 1168162, -3804809, 26747877, -14138091, 10609330, 12694420, 33473243, -13382104},
+ fieldElement{33184999, 11180355, 15832085, -11385430, -1633671, 225884, 15089336, -11023903, -6135662, 14480053},
+ },
+ {
+ fieldElement{31308717, -5619998, 31030840, -1897099, 15674547, -6582883, 5496208, 13685227, 27595050, 8737275},
+ fieldElement{-20318852, -15150239, 10933843, -16178022, 8335352, -7546022, -31008351, -12610604, 26498114, 66511},
+ fieldElement{22644454, -8761729, -16671776, 4884562, -3105614, -13559366, 30540766, -4286747, -13327787, -7515095},
+ },
+ {
+ fieldElement{-28017847, 9834845, 18617207, -2681312, -3401956, -13307506, 8205540, 13585437, -17127465, 15115439},
+ fieldElement{23711543, -672915, 31206561, -8362711, 6164647, -9709987, -33535882, -1426096, 8236921, 16492939},
+ fieldElement{-23910559, -13515526, -26299483, -4503841, 25005590, -7687270, 19574902, 10071562, 6708380, -6222424},
+ },
+ {
+ fieldElement{2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017, 9328700, 29955601, -11678310},
+ fieldElement{3096359, 9271816, -21620864, -15521844, -14847996, -7592937, -25892142, -12635595, -9917575, 6216608},
+ fieldElement{-32615849, 338663, -25195611, 2510422, -29213566, -13820213, 24822830, -6146567, -26767480, 7525079},
+ },
+ {
+ fieldElement{-23066649, -13985623, 16133487, -7896178, -3389565, 778788, -910336, -2782495, -19386633, 11994101},
+ fieldElement{21691500, -13624626, -641331, -14367021, 3285881, -3483596, -25064666, 9718258, -7477437, 13381418},
+ fieldElement{18445390, -4202236, 14979846, 11622458, -1727110, -3582980, 23111648, -6375247, 28535282, 15779576},
+ },
+ {
+ fieldElement{30098053, 3089662, -9234387, 16662135, -21306940, 11308411, -14068454, 12021730, 9955285, -16303356},
+ fieldElement{9734894, -14576830, -7473633, -9138735, 2060392, 11313496, -18426029, 9924399, 20194861, 13380996},
+ fieldElement{-26378102, -7965207, -22167821, 15789297, -18055342, -6168792, -1984914, 15707771, 26342023, 10146099},
+ },
+ },
+ {
+ {
+ fieldElement{-26016874, -219943, 21339191, -41388, 19745256, -2878700, -29637280, 2227040, 21612326, -545728},
+ fieldElement{-13077387, 1184228, 23562814, -5970442, -20351244, -6348714, 25764461, 12243797, -20856566, 11649658},
+ fieldElement{-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944, 6114064, 33514190, 2333242},
+ },
+ {
+ fieldElement{-21433588, -12421821, 8119782, 7219913, -21830522, -9016134, -6679750, -12670638, 24350578, -13450001},
+ fieldElement{-4116307, -11271533, -23886186, 4843615, -30088339, 690623, -31536088, -10406836, 8317860, 12352766},
+ fieldElement{18200138, -14475911, -33087759, -2696619, -23702521, -9102511, -23552096, -2287550, 20712163, 6719373},
+ },
+ {
+ fieldElement{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, -3763210, 26224235, -3297458},
+ fieldElement{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 21728352, 9493610, 18620611, -16428628},
+ fieldElement{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965, -5269471, -9725556, -30701573, -16479657},
+ },
+ {
+ fieldElement{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 12248509, -5240639, 13735342, 1934062},
+ fieldElement{25089769, 6742589, 17081145, -13406266, 21909293, -16067981, -15136294, -3765346, -21277997, 5473616},
+ fieldElement{31883677, -7961101, 1083432, -11572403, 22828471, 13290673, -7125085, 12469656, 29111212, -5451014},
+ },
+ {
+ fieldElement{24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 24367466, 6388839, -10295587, 452383},
+ fieldElement{-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, -24236251, -5915248, 15766062, 8407814},
+ fieldElement{-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, -8917023, -4388953, -8067909, 2276718},
+ },
+ {
+ fieldElement{30157918, 12924066, -17712050, 9245753, 19895028, 3368142, -23827587, 5096219, 22740376, -7303417},
+ fieldElement{2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 24051124, 13742383, -15637599, 13295222},
+ fieldElement{33338237, -8505733, 12532113, 7977527, 9106186, -1715251, -17720195, -4612972, -4451357, -14669444},
+ },
+ {
+ fieldElement{-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, -2469266, -4141880, 7770569, 9620597},
+ fieldElement{23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, -1694323, -33502340, -14767970},
+ fieldElement{1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 1220118, 30494170, -11440799},
+ },
+ {
+ fieldElement{-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, -26739026, 926050, -1684339, -13333647},
+ fieldElement{13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 9021034, 9078865, 3353509, 4033511},
+ fieldElement{-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 23161163, 8839127, 27485041, 7356032},
+ },
+ },
+ {
+ {
+ fieldElement{9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 2625015, 28431036, -16771834},
+ fieldElement{-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, -22545972, 14150565, 15970762, 4099461},
+ fieldElement{29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 13617293, -9937143, 11465739, 8317062},
+ },
+ {
+ fieldElement{-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 14898637, 3848455, 20969334, -5157516},
+ fieldElement{-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, -21610826, -3649888, 11177095, 14989547},
+ fieldElement{-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 13515641, 2581286, -28487508, 9930240},
+ },
+ {
+ fieldElement{-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 18345767, -13403753, 16291481, -5314038},
+ fieldElement{-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 6957617, 4368891, 9788741},
+ fieldElement{16660756, 7281060, -10830758, 12911820, 20108584, -8101676, -21722536, -8613148, 16250552, -11111103},
+ },
+ {
+ fieldElement{-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 10604807, -30190403, 4782747},
+ fieldElement{-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, -9981571, 4383045, 22546403, 437323},
+ fieldElement{31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 27343084, 2786261, -30633590, -14097016},
+ },
+ {
+ fieldElement{-14467279, -683715, -33374107, 7448552, 19294360, 14334329, -19690631, 2355319, -19284671, -6114373},
+ fieldElement{15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 18952177, 15496498, -29380133, 11754228},
+ fieldElement{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 7141596, 11724556, 22761615, -10134141},
+ },
+ {
+ fieldElement{16918416, 11729663, -18083579, 3022987, -31015732, -13339659, -28741185, -12227393, 32851222, 11717399},
+ fieldElement{11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 31474879, 3483633, -1193175, -4030831},
+ fieldElement{-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 33142652, 6546660, -19985279, -3948376},
+ },
+ {
+ fieldElement{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, -8537131, -12833048, -30772034, -15486313},
+ fieldElement{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, -31135347, -16049879, 10928917, 3011958},
+ fieldElement{-6957757, -15594337, 31696059, 334240, 29576716, 14796075, -30831056, -12805180, 18008031, 10258577},
+ },
+ {
+ fieldElement{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, -1853465, 1367120, 25127874, 6671743},
+ fieldElement{29701166, -14373934, -10878120, 9279288, -17568, 13127210, 21382910, 11042292, 25838796, 4642684},
+ fieldElement{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476},
+ },
+ },
+}
diff --git a/ed25519.go b/ed25519.go
new file mode 100644
index 0000000..9013eb3
--- /dev/null
+++ b/ed25519.go
@@ -0,0 +1,2242 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package ed25519 implements the Ed25519 signature algorithm. See
+// http://ed25519.cr.yp.to/.
+package ed25519
+
+// This code is a port of the public domain, "ref10" implementation of ed25519
+// from SUPERCOP.
+
+import (
+ "bytes"
+ "crypto/sha512"
+ "io"
+)
+
+const (
+ PublicKeySize = 32
+ PrivateKeySize = 64
+ SignatureSize = 64
+)
+
+// fieldElement represents an element of the field GF(2^255 - 19). An element
+// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
+// t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
+// context.
+type fieldElement [10]int32
+
+func feZero(fe *fieldElement) {
+ for i := range fe {
+ fe[i] = 0
+ }
+}
+
+func feOne(fe *fieldElement) {
+ feZero(fe)
+ fe[0] = 1
+}
+
+func feAdd(dst, a, b *fieldElement) {
+ for i := range dst {
+ dst[i] = a[i] + b[i]
+ }
+}
+
+func feSub(dst, a, b *fieldElement) {
+ for i := range dst {
+ dst[i] = a[i] - b[i]
+ }
+}
+
+func feCopy(dst, src *fieldElement) {
+ for i := range dst {
+ dst[i] = src[i]
+ }
+}
+
+// Replace (f,g) with (g,g) if b == 1;
+// replace (f,g) with (f,g) if b == 0.
+//
+// Preconditions: b in {0,1}.
+func feCMove(f, g *fieldElement, b int32) {
+ var x fieldElement
+ b = -b
+ for i := range x {
+ x[i] = b & (f[i] ^ g[i])
+ }
+
+ for i := range f {
+ f[i] ^= x[i]
+ }
+}
+
+func load3(in []byte) int64 {
+ var r int64
+ r = int64(in[0])
+ r |= int64(in[1]) << 8
+ r |= int64(in[2]) << 16
+ return r
+}
+
+func load4(in []byte) int64 {
+ var r int64
+ r = int64(in[0])
+ r |= int64(in[1]) << 8
+ r |= int64(in[2]) << 16
+ r |= int64(in[3]) << 24
+ return r
+}
+
+func feFromBytes(dst *fieldElement, src *[32]byte) {
+ h0 := load4(src[:])
+ h1 := load3(src[4:]) << 6
+ h2 := load3(src[7:]) << 5
+ h3 := load3(src[10:]) << 3
+ h4 := load3(src[13:]) << 2
+ h5 := load4(src[16:])
+ h6 := load3(src[20:]) << 7
+ h7 := load3(src[23:]) << 5
+ h8 := load3(src[26:]) << 4
+ h9 := (load3(src[29:]) & 8388607) << 2
+
+ var carry [10]int64
+ carry[9] = (h9 + 1<<24) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+ carry[1] = (h1 + 1<<24) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[3] = (h3 + 1<<24) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[5] = (h5 + 1<<24) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+ carry[7] = (h7 + 1<<24) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+
+ carry[0] = (h0 + 1<<25) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[2] = (h2 + 1<<25) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[4] = (h4 + 1<<25) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[6] = (h6 + 1<<25) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+ carry[8] = (h8 + 1<<25) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+
+ dst[0] = int32(h0)
+ dst[1] = int32(h1)
+ dst[2] = int32(h2)
+ dst[3] = int32(h3)
+ dst[4] = int32(h4)
+ dst[5] = int32(h5)
+ dst[6] = int32(h6)
+ dst[7] = int32(h7)
+ dst[8] = int32(h8)
+ dst[9] = int32(h9)
+}
+
+// feToBytes marshals h to s.
+// Preconditions:
+// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+//
+// Write p=2^255-19; q=floor(h/p).
+// Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
+//
+// Proof:
+// Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
+// Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4.
+//
+// Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
+// Then 0<y<1.
+//
+// Write r=h-pq.
+// Have 0<=r<=p-1=2^255-20.
+// Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
+//
+// Write x=r+19(2^-255)r+y.
+// Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
+//
+// Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
+// so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
+func feToBytes(s *[32]byte, h *fieldElement) {
+ var carry [10]int32
+
+ q := (19*h[9] + (1 << 24)) >> 25
+ q = (h[0] + q) >> 26
+ q = (h[1] + q) >> 25
+ q = (h[2] + q) >> 26
+ q = (h[3] + q) >> 25
+ q = (h[4] + q) >> 26
+ q = (h[5] + q) >> 25
+ q = (h[6] + q) >> 26
+ q = (h[7] + q) >> 25
+ q = (h[8] + q) >> 26
+ q = (h[9] + q) >> 25
+
+ // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20.
+ h[0] += 19 * q
+ // Goal: Output h-2^255 q, which is between 0 and 2^255-20.
+
+ carry[0] = h[0] >> 26
+ h[1] += carry[0]
+ h[0] -= carry[0] << 26
+ carry[1] = h[1] >> 25
+ h[2] += carry[1]
+ h[1] -= carry[1] << 25
+ carry[2] = h[2] >> 26
+ h[3] += carry[2]
+ h[2] -= carry[2] << 26
+ carry[3] = h[3] >> 25
+ h[4] += carry[3]
+ h[3] -= carry[3] << 25
+ carry[4] = h[4] >> 26
+ h[5] += carry[4]
+ h[4] -= carry[4] << 26
+ carry[5] = h[5] >> 25
+ h[6] += carry[5]
+ h[5] -= carry[5] << 25
+ carry[6] = h[6] >> 26
+ h[7] += carry[6]
+ h[6] -= carry[6] << 26
+ carry[7] = h[7] >> 25
+ h[8] += carry[7]
+ h[7] -= carry[7] << 25
+ carry[8] = h[8] >> 26
+ h[9] += carry[8]
+ h[8] -= carry[8] << 26
+ carry[9] = h[9] >> 25
+ h[9] -= carry[9] << 25
+ // h10 = carry9
+
+ // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
+ // Have h[0]+...+2^230 h[9] between 0 and 2^255-1;
+ // evidently 2^255 h10-2^255 q = 0.
+ // Goal: Output h[0]+...+2^230 h[9].
+
+ s[0] = byte(h[0] >> 0)
+ s[1] = byte(h[0] >> 8)
+ s[2] = byte(h[0] >> 16)
+ s[3] = byte((h[0] >> 24) | (h[1] << 2))
+ s[4] = byte(h[1] >> 6)
+ s[5] = byte(h[1] >> 14)
+ s[6] = byte((h[1] >> 22) | (h[2] << 3))
+ s[7] = byte(h[2] >> 5)
+ s[8] = byte(h[2] >> 13)
+ s[9] = byte((h[2] >> 21) | (h[3] << 5))
+ s[10] = byte(h[3] >> 3)
+ s[11] = byte(h[3] >> 11)
+ s[12] = byte((h[3] >> 19) | (h[4] << 6))
+ s[13] = byte(h[4] >> 2)
+ s[14] = byte(h[4] >> 10)
+ s[15] = byte(h[4] >> 18)
+ s[16] = byte(h[5] >> 0)
+ s[17] = byte(h[5] >> 8)
+ s[18] = byte(h[5] >> 16)
+ s[19] = byte((h[5] >> 24) | (h[6] << 1))
+ s[20] = byte(h[6] >> 7)
+ s[21] = byte(h[6] >> 15)
+ s[22] = byte((h[6] >> 23) | (h[7] << 3))
+ s[23] = byte(h[7] >> 5)
+ s[24] = byte(h[7] >> 13)
+ s[25] = byte((h[7] >> 21) | (h[8] << 4))
+ s[26] = byte(h[8] >> 4)
+ s[27] = byte(h[8] >> 12)
+ s[28] = byte((h[8] >> 20) | (h[9] << 6))
+ s[29] = byte(h[9] >> 2)
+ s[30] = byte(h[9] >> 10)
+ s[31] = byte(h[9] >> 18)
+}
+
+func feIsNegative(f *fieldElement) byte {
+ var s [32]byte
+ feToBytes(&s, f)
+ return s[0] & 1
+}
+
+func feIsNonZero(f *fieldElement) bool {
+ var s [32]byte
+ feToBytes(&s, f)
+ var x uint8
+ for _, b := range s {
+ x |= b
+ }
+ return x != 0
+}
+
+// feNeg sets h = -f
+//
+// Preconditions:
+// |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+//
+// Postconditions:
+// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+func feNeg(h, f *fieldElement) {
+ for i := range h {
+ h[i] = -f[i]
+ }
+}
+
+// feMul calculates h = f * g
+// Can overlap h with f or g.
+//
+// Preconditions:
+// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
+// |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
+//
+// Postconditions:
+// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+//
+// Notes on implementation strategy:
+//
+// Using schoolbook multiplication.
+// Karatsuba would save a little in some cost models.
+//
+// Most multiplications by 2 and 19 are 32-bit precomputations;
+// cheaper than 64-bit postcomputations.
+//
+// There is one remaining multiplication by 19 in the carry chain;
+// one *19 precomputation can be merged into this,
+// but the resulting data flow is considerably less clean.
+//
+// There are 12 carries below.
+// 10 of them are 2-way parallelizable and vectorizable.
+// Can get away with 11 carries, but then data flow is much deeper.
+//
+// With tighter constraints on inputs can squeeze carries into int32.
+func feMul(h, f, g *fieldElement) {
+ f0 := f[0]
+ f1 := f[1]
+ f2 := f[2]
+ f3 := f[3]
+ f4 := f[4]
+ f5 := f[5]
+ f6 := f[6]
+ f7 := f[7]
+ f8 := f[8]
+ f9 := f[9]
+ g0 := g[0]
+ g1 := g[1]
+ g2 := g[2]
+ g3 := g[3]
+ g4 := g[4]
+ g5 := g[5]
+ g6 := g[6]
+ g7 := g[7]
+ g8 := g[8]
+ g9 := g[9]
+ g1_19 := 19 * g1 /* 1.4*2^29 */
+ g2_19 := 19 * g2 /* 1.4*2^30; still ok */
+ g3_19 := 19 * g3
+ g4_19 := 19 * g4
+ g5_19 := 19 * g5
+ g6_19 := 19 * g6
+ g7_19 := 19 * g7
+ g8_19 := 19 * g8
+ g9_19 := 19 * g9
+ f1_2 := 2 * f1
+ f3_2 := 2 * f3
+ f5_2 := 2 * f5
+ f7_2 := 2 * f7
+ f9_2 := 2 * f9
+ f0g0 := int64(f0) * int64(g0)
+ f0g1 := int64(f0) * int64(g1)
+ f0g2 := int64(f0) * int64(g2)
+ f0g3 := int64(f0) * int64(g3)
+ f0g4 := int64(f0) * int64(g4)
+ f0g5 := int64(f0) * int64(g5)
+ f0g6 := int64(f0) * int64(g6)
+ f0g7 := int64(f0) * int64(g7)
+ f0g8 := int64(f0) * int64(g8)
+ f0g9 := int64(f0) * int64(g9)
+ f1g0 := int64(f1) * int64(g0)
+ f1g1_2 := int64(f1_2) * int64(g1)
+ f1g2 := int64(f1) * int64(g2)
+ f1g3_2 := int64(f1_2) * int64(g3)
+ f1g4 := int64(f1) * int64(g4)
+ f1g5_2 := int64(f1_2) * int64(g5)
+ f1g6 := int64(f1) * int64(g6)
+ f1g7_2 := int64(f1_2) * int64(g7)
+ f1g8 := int64(f1) * int64(g8)
+ f1g9_38 := int64(f1_2) * int64(g9_19)
+ f2g0 := int64(f2) * int64(g0)
+ f2g1 := int64(f2) * int64(g1)
+ f2g2 := int64(f2) * int64(g2)
+ f2g3 := int64(f2) * int64(g3)
+ f2g4 := int64(f2) * int64(g4)
+ f2g5 := int64(f2) * int64(g5)
+ f2g6 := int64(f2) * int64(g6)
+ f2g7 := int64(f2) * int64(g7)
+ f2g8_19 := int64(f2) * int64(g8_19)
+ f2g9_19 := int64(f2) * int64(g9_19)
+ f3g0 := int64(f3) * int64(g0)
+ f3g1_2 := int64(f3_2) * int64(g1)
+ f3g2 := int64(f3) * int64(g2)
+ f3g3_2 := int64(f3_2) * int64(g3)
+ f3g4 := int64(f3) * int64(g4)
+ f3g5_2 := int64(f3_2) * int64(g5)
+ f3g6 := int64(f3) * int64(g6)
+ f3g7_38 := int64(f3_2) * int64(g7_19)
+ f3g8_19 := int64(f3) * int64(g8_19)
+ f3g9_38 := int64(f3_2) * int64(g9_19)
+ f4g0 := int64(f4) * int64(g0)
+ f4g1 := int64(f4) * int64(g1)
+ f4g2 := int64(f4) * int64(g2)
+ f4g3 := int64(f4) * int64(g3)
+ f4g4 := int64(f4) * int64(g4)
+ f4g5 := int64(f4) * int64(g5)
+ f4g6_19 := int64(f4) * int64(g6_19)
+ f4g7_19 := int64(f4) * int64(g7_19)
+ f4g8_19 := int64(f4) * int64(g8_19)
+ f4g9_19 := int64(f4) * int64(g9_19)
+ f5g0 := int64(f5) * int64(g0)
+ f5g1_2 := int64(f5_2) * int64(g1)
+ f5g2 := int64(f5) * int64(g2)
+ f5g3_2 := int64(f5_2) * int64(g3)
+ f5g4 := int64(f5) * int64(g4)
+ f5g5_38 := int64(f5_2) * int64(g5_19)
+ f5g6_19 := int64(f5) * int64(g6_19)
+ f5g7_38 := int64(f5_2) * int64(g7_19)
+ f5g8_19 := int64(f5) * int64(g8_19)
+ f5g9_38 := int64(f5_2) * int64(g9_19)
+ f6g0 := int64(f6) * int64(g0)
+ f6g1 := int64(f6) * int64(g1)
+ f6g2 := int64(f6) * int64(g2)
+ f6g3 := int64(f6) * int64(g3)
+ f6g4_19 := int64(f6) * int64(g4_19)
+ f6g5_19 := int64(f6) * int64(g5_19)
+ f6g6_19 := int64(f6) * int64(g6_19)
+ f6g7_19 := int64(f6) * int64(g7_19)
+ f6g8_19 := int64(f6) * int64(g8_19)
+ f6g9_19 := int64(f6) * int64(g9_19)
+ f7g0 := int64(f7) * int64(g0)
+ f7g1_2 := int64(f7_2) * int64(g1)
+ f7g2 := int64(f7) * int64(g2)
+ f7g3_38 := int64(f7_2) * int64(g3_19)
+ f7g4_19 := int64(f7) * int64(g4_19)
+ f7g5_38 := int64(f7_2) * int64(g5_19)
+ f7g6_19 := int64(f7) * int64(g6_19)
+ f7g7_38 := int64(f7_2) * int64(g7_19)
+ f7g8_19 := int64(f7) * int64(g8_19)
+ f7g9_38 := int64(f7_2) * int64(g9_19)
+ f8g0 := int64(f8) * int64(g0)
+ f8g1 := int64(f8) * int64(g1)
+ f8g2_19 := int64(f8) * int64(g2_19)
+ f8g3_19 := int64(f8) * int64(g3_19)
+ f8g4_19 := int64(f8) * int64(g4_19)
+ f8g5_19 := int64(f8) * int64(g5_19)
+ f8g6_19 := int64(f8) * int64(g6_19)
+ f8g7_19 := int64(f8) * int64(g7_19)
+ f8g8_19 := int64(f8) * int64(g8_19)
+ f8g9_19 := int64(f8) * int64(g9_19)
+ f9g0 := int64(f9) * int64(g0)
+ f9g1_38 := int64(f9_2) * int64(g1_19)
+ f9g2_19 := int64(f9) * int64(g2_19)
+ f9g3_38 := int64(f9_2) * int64(g3_19)
+ f9g4_19 := int64(f9) * int64(g4_19)
+ f9g5_38 := int64(f9_2) * int64(g5_19)
+ f9g6_19 := int64(f9) * int64(g6_19)
+ f9g7_38 := int64(f9_2) * int64(g7_19)
+ f9g8_19 := int64(f9) * int64(g8_19)
+ f9g9_38 := int64(f9_2) * int64(g9_19)
+ h0 := f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38
+ h1 := f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19
+ h2 := f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38
+ h3 := f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19
+ h4 := f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38
+ h5 := f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19
+ h6 := f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38
+ h7 := f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19
+ h8 := f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38
+ h9 := f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0
+ var carry [10]int64
+
+ /*
+ |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38))
+ i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8
+ |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19))
+ i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9
+ */
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ /* |h0| <= 2^25 */
+ /* |h4| <= 2^25 */
+ /* |h1| <= 1.51*2^58 */
+ /* |h5| <= 1.51*2^58 */
+
+ carry[1] = (h1 + (1 << 24)) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[5] = (h5 + (1 << 24)) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+ /* |h1| <= 2^24; from now on fits into int32 */
+ /* |h5| <= 2^24; from now on fits into int32 */
+ /* |h2| <= 1.21*2^59 */
+ /* |h6| <= 1.21*2^59 */
+
+ carry[2] = (h2 + (1 << 25)) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[6] = (h6 + (1 << 25)) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+ /* |h2| <= 2^25; from now on fits into int32 unchanged */
+ /* |h6| <= 2^25; from now on fits into int32 unchanged */
+ /* |h3| <= 1.51*2^58 */
+ /* |h7| <= 1.51*2^58 */
+
+ carry[3] = (h3 + (1 << 24)) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[7] = (h7 + (1 << 24)) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+ /* |h3| <= 2^24; from now on fits into int32 unchanged */
+ /* |h7| <= 2^24; from now on fits into int32 unchanged */
+ /* |h4| <= 1.52*2^33 */
+ /* |h8| <= 1.52*2^33 */
+
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[8] = (h8 + (1 << 25)) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+ /* |h4| <= 2^25; from now on fits into int32 unchanged */
+ /* |h8| <= 2^25; from now on fits into int32 unchanged */
+ /* |h5| <= 1.01*2^24 */
+ /* |h9| <= 1.51*2^58 */
+
+ carry[9] = (h9 + (1 << 24)) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+ /* |h9| <= 2^24; from now on fits into int32 unchanged */
+ /* |h0| <= 1.8*2^37 */
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ /* |h0| <= 2^25; from now on fits into int32 unchanged */
+ /* |h1| <= 1.01*2^24 */
+
+ h[0] = int32(h0)
+ h[1] = int32(h1)
+ h[2] = int32(h2)
+ h[3] = int32(h3)
+ h[4] = int32(h4)
+ h[5] = int32(h5)
+ h[6] = int32(h6)
+ h[7] = int32(h7)
+ h[8] = int32(h8)
+ h[9] = int32(h9)
+}
+
+// feSquare calculates h = f*f. Can overlap h with f.
+//
+// Preconditions:
+// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
+//
+// Postconditions:
+// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+func feSquare(h, f *fieldElement) {
+ f0 := f[0]
+ f1 := f[1]
+ f2 := f[2]
+ f3 := f[3]
+ f4 := f[4]
+ f5 := f[5]
+ f6 := f[6]
+ f7 := f[7]
+ f8 := f[8]
+ f9 := f[9]
+ f0_2 := 2 * f0
+ f1_2 := 2 * f1
+ f2_2 := 2 * f2
+ f3_2 := 2 * f3
+ f4_2 := 2 * f4
+ f5_2 := 2 * f5
+ f6_2 := 2 * f6
+ f7_2 := 2 * f7
+ f5_38 := 38 * f5 // 1.31*2^30
+ f6_19 := 19 * f6 // 1.31*2^30
+ f7_38 := 38 * f7 // 1.31*2^30
+ f8_19 := 19 * f8 // 1.31*2^30
+ f9_38 := 38 * f9 // 1.31*2^30
+ f0f0 := int64(f0) * int64(f0)
+ f0f1_2 := int64(f0_2) * int64(f1)
+ f0f2_2 := int64(f0_2) * int64(f2)
+ f0f3_2 := int64(f0_2) * int64(f3)
+ f0f4_2 := int64(f0_2) * int64(f4)
+ f0f5_2 := int64(f0_2) * int64(f5)
+ f0f6_2 := int64(f0_2) * int64(f6)
+ f0f7_2 := int64(f0_2) * int64(f7)
+ f0f8_2 := int64(f0_2) * int64(f8)
+ f0f9_2 := int64(f0_2) * int64(f9)
+ f1f1_2 := int64(f1_2) * int64(f1)
+ f1f2_2 := int64(f1_2) * int64(f2)
+ f1f3_4 := int64(f1_2) * int64(f3_2)
+ f1f4_2 := int64(f1_2) * int64(f4)
+ f1f5_4 := int64(f1_2) * int64(f5_2)
+ f1f6_2 := int64(f1_2) * int64(f6)
+ f1f7_4 := int64(f1_2) * int64(f7_2)
+ f1f8_2 := int64(f1_2) * int64(f8)
+ f1f9_76 := int64(f1_2) * int64(f9_38)
+ f2f2 := int64(f2) * int64(f2)
+ f2f3_2 := int64(f2_2) * int64(f3)
+ f2f4_2 := int64(f2_2) * int64(f4)
+ f2f5_2 := int64(f2_2) * int64(f5)
+ f2f6_2 := int64(f2_2) * int64(f6)
+ f2f7_2 := int64(f2_2) * int64(f7)
+ f2f8_38 := int64(f2_2) * int64(f8_19)
+ f2f9_38 := int64(f2) * int64(f9_38)
+ f3f3_2 := int64(f3_2) * int64(f3)
+ f3f4_2 := int64(f3_2) * int64(f4)
+ f3f5_4 := int64(f3_2) * int64(f5_2)
+ f3f6_2 := int64(f3_2) * int64(f6)
+ f3f7_76 := int64(f3_2) * int64(f7_38)
+ f3f8_38 := int64(f3_2) * int64(f8_19)
+ f3f9_76 := int64(f3_2) * int64(f9_38)
+ f4f4 := int64(f4) * int64(f4)
+ f4f5_2 := int64(f4_2) * int64(f5)
+ f4f6_38 := int64(f4_2) * int64(f6_19)
+ f4f7_38 := int64(f4) * int64(f7_38)
+ f4f8_38 := int64(f4_2) * int64(f8_19)
+ f4f9_38 := int64(f4) * int64(f9_38)
+ f5f5_38 := int64(f5) * int64(f5_38)
+ f5f6_38 := int64(f5_2) * int64(f6_19)
+ f5f7_76 := int64(f5_2) * int64(f7_38)
+ f5f8_38 := int64(f5_2) * int64(f8_19)
+ f5f9_76 := int64(f5_2) * int64(f9_38)
+ f6f6_19 := int64(f6) * int64(f6_19)
+ f6f7_38 := int64(f6) * int64(f7_38)
+ f6f8_38 := int64(f6_2) * int64(f8_19)
+ f6f9_38 := int64(f6) * int64(f9_38)
+ f7f7_38 := int64(f7) * int64(f7_38)
+ f7f8_38 := int64(f7_2) * int64(f8_19)
+ f7f9_76 := int64(f7_2) * int64(f9_38)
+ f8f8_19 := int64(f8) * int64(f8_19)
+ f8f9_38 := int64(f8) * int64(f9_38)
+ f9f9_38 := int64(f9) * int64(f9_38)
+ h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38
+ h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38
+ h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19
+ h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38
+ h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38
+ h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38
+ h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19
+ h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38
+ h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38
+ h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2
+ var carry [10]int64
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+
+ carry[1] = (h1 + (1 << 24)) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[5] = (h5 + (1 << 24)) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+
+ carry[2] = (h2 + (1 << 25)) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[6] = (h6 + (1 << 25)) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+
+ carry[3] = (h3 + (1 << 24)) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[7] = (h7 + (1 << 24)) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[8] = (h8 + (1 << 25)) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+
+ carry[9] = (h9 + (1 << 24)) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+
+ h[0] = int32(h0)
+ h[1] = int32(h1)
+ h[2] = int32(h2)
+ h[3] = int32(h3)
+ h[4] = int32(h4)
+ h[5] = int32(h5)
+ h[6] = int32(h6)
+ h[7] = int32(h7)
+ h[8] = int32(h8)
+ h[9] = int32(h9)
+}
+
+// feSquare2 sets h = 2 * f * f
+//
+// Can overlap h with f.
+//
+// Preconditions:
+// |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
+//
+// Postconditions:
+// |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
+// See fe_mul.c for discussion of implementation strategy.
+func feSquare2(h, f *fieldElement) {
+ f0 := f[0]
+ f1 := f[1]
+ f2 := f[2]
+ f3 := f[3]
+ f4 := f[4]
+ f5 := f[5]
+ f6 := f[6]
+ f7 := f[7]
+ f8 := f[8]
+ f9 := f[9]
+ f0_2 := 2 * f0
+ f1_2 := 2 * f1
+ f2_2 := 2 * f2
+ f3_2 := 2 * f3
+ f4_2 := 2 * f4
+ f5_2 := 2 * f5
+ f6_2 := 2 * f6
+ f7_2 := 2 * f7
+ f5_38 := 38 * f5 // 1.959375*2^30
+ f6_19 := 19 * f6 // 1.959375*2^30
+ f7_38 := 38 * f7 // 1.959375*2^30
+ f8_19 := 19 * f8 // 1.959375*2^30
+ f9_38 := 38 * f9 // 1.959375*2^30
+ f0f0 := int64(f0) * int64(f0)
+ f0f1_2 := int64(f0_2) * int64(f1)
+ f0f2_2 := int64(f0_2) * int64(f2)
+ f0f3_2 := int64(f0_2) * int64(f3)
+ f0f4_2 := int64(f0_2) * int64(f4)
+ f0f5_2 := int64(f0_2) * int64(f5)
+ f0f6_2 := int64(f0_2) * int64(f6)
+ f0f7_2 := int64(f0_2) * int64(f7)
+ f0f8_2 := int64(f0_2) * int64(f8)
+ f0f9_2 := int64(f0_2) * int64(f9)
+ f1f1_2 := int64(f1_2) * int64(f1)
+ f1f2_2 := int64(f1_2) * int64(f2)
+ f1f3_4 := int64(f1_2) * int64(f3_2)
+ f1f4_2 := int64(f1_2) * int64(f4)
+ f1f5_4 := int64(f1_2) * int64(f5_2)
+ f1f6_2 := int64(f1_2) * int64(f6)
+ f1f7_4 := int64(f1_2) * int64(f7_2)
+ f1f8_2 := int64(f1_2) * int64(f8)
+ f1f9_76 := int64(f1_2) * int64(f9_38)
+ f2f2 := int64(f2) * int64(f2)
+ f2f3_2 := int64(f2_2) * int64(f3)
+ f2f4_2 := int64(f2_2) * int64(f4)
+ f2f5_2 := int64(f2_2) * int64(f5)
+ f2f6_2 := int64(f2_2) * int64(f6)
+ f2f7_2 := int64(f2_2) * int64(f7)
+ f2f8_38 := int64(f2_2) * int64(f8_19)
+ f2f9_38 := int64(f2) * int64(f9_38)
+ f3f3_2 := int64(f3_2) * int64(f3)
+ f3f4_2 := int64(f3_2) * int64(f4)
+ f3f5_4 := int64(f3_2) * int64(f5_2)
+ f3f6_2 := int64(f3_2) * int64(f6)
+ f3f7_76 := int64(f3_2) * int64(f7_38)
+ f3f8_38 := int64(f3_2) * int64(f8_19)
+ f3f9_76 := int64(f3_2) * int64(f9_38)
+ f4f4 := int64(f4) * int64(f4)
+ f4f5_2 := int64(f4_2) * int64(f5)
+ f4f6_38 := int64(f4_2) * int64(f6_19)
+ f4f7_38 := int64(f4) * int64(f7_38)
+ f4f8_38 := int64(f4_2) * int64(f8_19)
+ f4f9_38 := int64(f4) * int64(f9_38)
+ f5f5_38 := int64(f5) * int64(f5_38)
+ f5f6_38 := int64(f5_2) * int64(f6_19)
+ f5f7_76 := int64(f5_2) * int64(f7_38)
+ f5f8_38 := int64(f5_2) * int64(f8_19)
+ f5f9_76 := int64(f5_2) * int64(f9_38)
+ f6f6_19 := int64(f6) * int64(f6_19)
+ f6f7_38 := int64(f6) * int64(f7_38)
+ f6f8_38 := int64(f6_2) * int64(f8_19)
+ f6f9_38 := int64(f6) * int64(f9_38)
+ f7f7_38 := int64(f7) * int64(f7_38)
+ f7f8_38 := int64(f7_2) * int64(f8_19)
+ f7f9_76 := int64(f7_2) * int64(f9_38)
+ f8f8_19 := int64(f8) * int64(f8_19)
+ f8f9_38 := int64(f8) * int64(f9_38)
+ f9f9_38 := int64(f9) * int64(f9_38)
+ h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38
+ h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38
+ h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19
+ h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38
+ h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38
+ h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38
+ h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19
+ h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38
+ h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38
+ h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2
+ var carry [10]int64
+
+ h0 += h0
+ h1 += h1
+ h2 += h2
+ h3 += h3
+ h4 += h4
+ h5 += h5
+ h6 += h6
+ h7 += h7
+ h8 += h8
+ h9 += h9
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+
+ carry[1] = (h1 + (1 << 24)) >> 25
+ h2 += carry[1]
+ h1 -= carry[1] << 25
+ carry[5] = (h5 + (1 << 24)) >> 25
+ h6 += carry[5]
+ h5 -= carry[5] << 25
+
+ carry[2] = (h2 + (1 << 25)) >> 26
+ h3 += carry[2]
+ h2 -= carry[2] << 26
+ carry[6] = (h6 + (1 << 25)) >> 26
+ h7 += carry[6]
+ h6 -= carry[6] << 26
+
+ carry[3] = (h3 + (1 << 24)) >> 25
+ h4 += carry[3]
+ h3 -= carry[3] << 25
+ carry[7] = (h7 + (1 << 24)) >> 25
+ h8 += carry[7]
+ h7 -= carry[7] << 25
+
+ carry[4] = (h4 + (1 << 25)) >> 26
+ h5 += carry[4]
+ h4 -= carry[4] << 26
+ carry[8] = (h8 + (1 << 25)) >> 26
+ h9 += carry[8]
+ h8 -= carry[8] << 26
+
+ carry[9] = (h9 + (1 << 24)) >> 25
+ h0 += carry[9] * 19
+ h9 -= carry[9] << 25
+
+ carry[0] = (h0 + (1 << 25)) >> 26
+ h1 += carry[0]
+ h0 -= carry[0] << 26
+
+ h[0] = int32(h0)
+ h[1] = int32(h1)
+ h[2] = int32(h2)
+ h[3] = int32(h3)
+ h[4] = int32(h4)
+ h[5] = int32(h5)
+ h[6] = int32(h6)
+ h[7] = int32(h7)
+ h[8] = int32(h8)
+ h[9] = int32(h9)
+}
+
+func feInvert(out, z *fieldElement) {
+ var t0, t1, t2, t3 fieldElement
+ var i int
+
+ feSquare(&t0, z)
+ for i = 1; i < 1; i++ {
+ feSquare(&t0, &t0)
+ }
+ feSquare(&t1, &t0)
+ for i = 1; i < 2; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t1, z, &t1)
+ feMul(&t0, &t0, &t1)
+ feSquare(&t2, &t0)
+ for i = 1; i < 1; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t1, &t2)
+ feSquare(&t2, &t1)
+ for i = 1; i < 5; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t2, &t1)
+ for i = 1; i < 10; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t2, &t2, &t1)
+ feSquare(&t3, &t2)
+ for i = 1; i < 20; i++ {
+ feSquare(&t3, &t3)
+ }
+ feMul(&t2, &t3, &t2)
+ feSquare(&t2, &t2)
+ for i = 1; i < 10; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t2, &t1)
+ for i = 1; i < 50; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t2, &t2, &t1)
+ feSquare(&t3, &t2)
+ for i = 1; i < 100; i++ {
+ feSquare(&t3, &t3)
+ }
+ feMul(&t2, &t3, &t2)
+ feSquare(&t2, &t2)
+ for i = 1; i < 50; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t1, &t1)
+ for i = 1; i < 5; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(out, &t1, &t0)
+}
+
+func fePow22523(out, z *fieldElement) {
+ var t0, t1, t2 fieldElement
+ var i int
+
+ feSquare(&t0, z)
+ for i = 1; i < 1; i++ {
+ feSquare(&t0, &t0)
+ }
+ feSquare(&t1, &t0)
+ for i = 1; i < 2; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t1, z, &t1)
+ feMul(&t0, &t0, &t1)
+ feSquare(&t0, &t0)
+ for i = 1; i < 1; i++ {
+ feSquare(&t0, &t0)
+ }
+ feMul(&t0, &t1, &t0)
+ feSquare(&t1, &t0)
+ for i = 1; i < 5; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t0, &t1, &t0)
+ feSquare(&t1, &t0)
+ for i = 1; i < 10; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t1, &t1, &t0)
+ feSquare(&t2, &t1)
+ for i = 1; i < 20; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t1, &t1)
+ for i = 1; i < 10; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t0, &t1, &t0)
+ feSquare(&t1, &t0)
+ for i = 1; i < 50; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t1, &t1, &t0)
+ feSquare(&t2, &t1)
+ for i = 1; i < 100; i++ {
+ feSquare(&t2, &t2)
+ }
+ feMul(&t1, &t2, &t1)
+ feSquare(&t1, &t1)
+ for i = 1; i < 50; i++ {
+ feSquare(&t1, &t1)
+ }
+ feMul(&t0, &t1, &t0)
+ feSquare(&t0, &t0)
+ for i = 1; i < 2; i++ {
+ feSquare(&t0, &t0)
+ }
+ feMul(out, &t0, z)
+}
+
+// Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 *
+// y^2 where d = -121665/121666.
+//
+// Several representations are used:
+// projectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z
+// extendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
+// completedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
+// preComputedGroupElement: (y+x,y-x,2dxy)
+
+type projectiveGroupElement struct {
+ X, Y, Z fieldElement
+}
+
+type extendedGroupElement struct {
+ X, Y, Z, T fieldElement
+}
+
+type completedGroupElement struct {
+ X, Y, Z, T fieldElement
+}
+
+type preComputedGroupElement struct {
+ yPlusX, yMinusX, xy2d fieldElement
+}
+
+type cachedGroupElement struct {
+ yPlusX, yMinusX, Z, T2d fieldElement
+}
+
+func (p *projectiveGroupElement) Zero() {
+ feZero(&p.X)
+ feOne(&p.Y)
+ feOne(&p.Z)
+}
+
+func (p *projectiveGroupElement) Double(r *completedGroupElement) {
+ var t0 fieldElement
+
+ feSquare(&r.X, &p.X)
+ feSquare(&r.Z, &p.Y)
+ feSquare2(&r.T, &p.Z)
+ feAdd(&r.Y, &p.X, &p.Y)
+ feSquare(&t0, &r.Y)
+ feAdd(&r.Y, &r.Z, &r.X)
+ feSub(&r.Z, &r.Z, &r.X)
+ feSub(&r.X, &t0, &r.Y)
+ feSub(&r.T, &r.T, &r.Z)
+}
+
+func (p *projectiveGroupElement) ToBytes(s *[32]byte) {
+ var recip, x, y fieldElement
+
+ feInvert(&recip, &p.Z)
+ feMul(&x, &p.X, &recip)
+ feMul(&y, &p.Y, &recip)
+ feToBytes(s, &y)
+ s[31] ^= feIsNegative(&x) << 7
+}
+
+func (p *extendedGroupElement) Zero() {
+ feZero(&p.X)
+ feOne(&p.Y)
+ feOne(&p.Z)
+ feZero(&p.T)
+}
+
+func (p *extendedGroupElement) Double(r *completedGroupElement) {
+ var q projectiveGroupElement
+ p.ToProjective(&q)
+ q.Double(r)
+}
+
+func (p *extendedGroupElement) ToCached(r *cachedGroupElement) {
+ feAdd(&r.yPlusX, &p.Y, &p.X)
+ feSub(&r.yMinusX, &p.Y, &p.X)
+ feCopy(&r.Z, &p.Z)
+ feMul(&r.T2d, &p.T, &d2)
+}
+
+func (p *extendedGroupElement) ToProjective(r *projectiveGroupElement) {
+ feCopy(&r.X, &p.X)
+ feCopy(&r.Y, &p.Y)
+ feCopy(&r.Z, &p.Z)
+}
+
+func (p *extendedGroupElement) ToBytes(s *[32]byte) {
+ var recip, x, y fieldElement
+
+ feInvert(&recip, &p.Z)
+ feMul(&x, &p.X, &recip)
+ feMul(&y, &p.Y, &recip)
+ feToBytes(s, &y)
+ s[31] ^= feIsNegative(&x) << 7
+}
+
+func (p *extendedGroupElement) FromBytes(s *[32]byte) bool {
+ var u, v, v3, vxx, check fieldElement
+
+ feFromBytes(&p.Y, s)
+ feOne(&p.Z)
+ feSquare(&u, &p.Y)
+ feMul(&v, &u, &d)
+ feSub(&u, &u, &p.Z) // y = y^2-1
+ feAdd(&v, &v, &p.Z) // v = dy^2+1
+
+ feSquare(&v3, &v)
+ feMul(&v3, &v3, &v) // v3 = v^3
+ feSquare(&p.X, &v3)
+ feMul(&p.X, &p.X, &v)
+ feMul(&p.X, &p.X, &u) // x = uv^7
+
+ fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8)
+ feMul(&p.X, &p.X, &v3)
+ feMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8)
+
+ var tmpX, tmp2 [32]byte
+
+ feSquare(&vxx, &p.X)
+ feMul(&vxx, &vxx, &v)
+ feSub(&check, &vxx, &u) // vx^2-u
+ if feIsNonZero(&check) {
+ feAdd(&check, &vxx, &u) // vx^2+u
+ if feIsNonZero(&check) {
+ return false
+ }
+ feMul(&p.X, &p.X, &sqrtm1)
+
+ feToBytes(&tmpX, &p.X)
+ for i, v := range tmpX {
+ tmp2[31-i] = v
+ }
+ }
+
+ if feIsNegative(&p.X) == (s[31] >> 7) {
+ feNeg(&p.X, &p.X)
+ }
+
+ feMul(&p.T, &p.X, &p.Y)
+ return true
+}
+
+func (p *completedGroupElement) ToProjective(r *projectiveGroupElement) {
+ feMul(&r.X, &p.X, &p.T)
+ feMul(&r.Y, &p.Y, &p.Z)
+ feMul(&r.Z, &p.Z, &p.T)
+}
+
+func (p *completedGroupElement) ToExtended(r *extendedGroupElement) {
+ feMul(&r.X, &p.X, &p.T)
+ feMul(&r.Y, &p.Y, &p.Z)
+ feMul(&r.Z, &p.Z, &p.T)
+ feMul(&r.T, &p.X, &p.Y)
+}
+
+func (p *preComputedGroupElement) Zero() {
+ feOne(&p.yPlusX)
+ feOne(&p.yMinusX)
+ feZero(&p.xy2d)
+}
+
+func geAdd(r *completedGroupElement, p *extendedGroupElement, q *cachedGroupElement) {
+ var t0 fieldElement
+
+ feAdd(&r.X, &p.Y, &p.X)
+ feSub(&r.Y, &p.Y, &p.X)
+ feMul(&r.Z, &r.X, &q.yPlusX)
+ feMul(&r.Y, &r.Y, &q.yMinusX)
+ feMul(&r.T, &q.T2d, &p.T)
+ feMul(&r.X, &p.Z, &q.Z)
+ feAdd(&t0, &r.X, &r.X)
+ feSub(&r.X, &r.Z, &r.Y)
+ feAdd(&r.Y, &r.Z, &r.Y)
+ feAdd(&r.Z, &t0, &r.T)
+ feSub(&r.T, &t0, &r.T)
+}
+
+func geSub(r *completedGroupElement, p *extendedGroupElement, q *cachedGroupElement) {
+ var t0 fieldElement
+
+ feAdd(&r.X, &p.Y, &p.X)
+ feSub(&r.Y, &p.Y, &p.X)
+ feMul(&r.Z, &r.X, &q.yMinusX)
+ feMul(&r.Y, &r.Y, &q.yPlusX)
+ feMul(&r.T, &q.T2d, &p.T)
+ feMul(&r.X, &p.Z, &q.Z)
+ feAdd(&t0, &r.X, &r.X)
+ feSub(&r.X, &r.Z, &r.Y)
+ feAdd(&r.Y, &r.Z, &r.Y)
+ feSub(&r.Z, &t0, &r.T)
+ feAdd(&r.T, &t0, &r.T)
+}
+
+func geMixedAdd(r *completedGroupElement, p *extendedGroupElement, q *preComputedGroupElement) {
+ var t0 fieldElement
+
+ feAdd(&r.X, &p.Y, &p.X)
+ feSub(&r.Y, &p.Y, &p.X)
+ feMul(&r.Z, &r.X, &q.yPlusX)
+ feMul(&r.Y, &r.Y, &q.yMinusX)
+ feMul(&r.T, &q.xy2d, &p.T)
+ feAdd(&t0, &p.Z, &p.Z)
+ feSub(&r.X, &r.Z, &r.Y)
+ feAdd(&r.Y, &r.Z, &r.Y)
+ feAdd(&r.Z, &t0, &r.T)
+ feSub(&r.T, &t0, &r.T)
+}
+
+func geMixedSub(r *completedGroupElement, p *extendedGroupElement, q *preComputedGroupElement) {
+ var t0 fieldElement
+
+ feAdd(&r.X, &p.Y, &p.X)
+ feSub(&r.Y, &p.Y, &p.X)
+ feMul(&r.Z, &r.X, &q.yMinusX)
+ feMul(&r.Y, &r.Y, &q.yPlusX)
+ feMul(&r.T, &q.xy2d, &p.T)
+ feAdd(&t0, &p.Z, &p.Z)
+ feSub(&r.X, &r.Z, &r.Y)
+ feAdd(&r.Y, &r.Z, &r.Y)
+ feSub(&r.Z, &t0, &r.T)
+ feAdd(&r.T, &t0, &r.T)
+}
+
+func slide(r *[256]int8, a *[32]byte) {
+ for i := range r {
+ r[i] = int8(1 & (a[i>>3] >> uint(i&7)))
+ }
+
+ for i := range r {
+ if r[i] != 0 {
+ for b := 1; b <= 6 && i+b < 256; b++ {
+ if r[i+b] != 0 {
+ if r[i]+(r[i+b]<<uint(b)) <= 15 {
+ r[i] += r[i+b] << uint(b)
+ r[i+b] = 0
+ } else if r[i]-(r[i+b]<<uint(b)) >= -15 {
+ r[i] -= r[i+b] << uint(b)
+ for k := i + b; k < 256; k++ {
+ if r[k] == 0 {
+ r[k] = 1
+ break
+ }
+ r[k] = 0
+ }
+ } else {
+ break
+ }
+ }
+ }
+ }
+ }
+}
+
+// geDoubleScalarMultVartime sets r = a*A + b*B
+// where a = a[0]+256*a[1]+...+256^31 a[31].
+// and b = b[0]+256*b[1]+...+256^31 b[31].
+// B is the Ed25519 base point (x,4/5) with x positive.
+func geDoubleScalarMultVartime(r *projectiveGroupElement, a *[32]byte, A *extendedGroupElement, b *[32]byte) {
+ var aSlide, bSlide [256]int8
+ var Ai [8]cachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A
+ var t completedGroupElement
+ var u, A2 extendedGroupElement
+ var i int
+
+ slide(&aSlide, a)
+ slide(&bSlide, b)
+
+ A.ToCached(&Ai[0])
+ A.Double(&t)
+ t.ToExtended(&A2)
+
+ for i := 0; i < 7; i++ {
+ geAdd(&t, &A2, &Ai[i])
+ t.ToExtended(&u)
+ u.ToCached(&Ai[i+1])
+ }
+
+ r.Zero()
+
+ for i = 255; i >= 0; i-- {
+ if aSlide[i] != 0 || bSlide[i] != 0 {
+ break
+ }
+ }
+
+ for ; i >= 0; i-- {
+ r.Double(&t)
+
+ if aSlide[i] > 0 {
+ t.ToExtended(&u)
+ geAdd(&t, &u, &Ai[aSlide[i]/2])
+ } else if aSlide[i] < 0 {
+ t.ToExtended(&u)
+ geSub(&t, &u, &Ai[(-aSlide[i])/2])
+ }
+
+ if bSlide[i] > 0 {
+ t.ToExtended(&u)
+ geMixedAdd(&t, &u, &bi[bSlide[i]/2])
+ } else if bSlide[i] < 0 {
+ t.ToExtended(&u)
+ geMixedSub(&t, &u, &bi[(-bSlide[i])/2])
+ }
+
+ t.ToProjective(r)
+ }
+}
+
+// equal returns 1 if b == c and 0 otherwise.
+func equal(b, c int32) int32 {
+ x := uint32(b ^ c)
+ x--
+ return int32(x >> 31)
+}
+
+// negative returns 1 if b < 0 and 0 otherwise.
+func negative(b int32) int32 {
+ return (b >> 31) & 1
+}
+
+func preComputedGroupElementCMove(t, u *preComputedGroupElement, b int32) {
+ feCMove(&t.yPlusX, &u.yPlusX, b)
+ feCMove(&t.yMinusX, &u.yMinusX, b)
+ feCMove(&t.xy2d, &u.xy2d, b)
+}
+
+func selectPoint(t *preComputedGroupElement, pos int32, b int32) {
+ var minusT preComputedGroupElement
+ bNegative := negative(b)
+ bAbs := b - (((-bNegative) & b) << 1)
+
+ t.Zero()
+ for i := int32(0); i < 8; i++ {
+ preComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1))
+ }
+ feCopy(&minusT.yPlusX, &t.yMinusX)
+ feCopy(&minusT.yMinusX, &t.yPlusX)
+ feNeg(&minusT.xy2d, &t.xy2d)
+ preComputedGroupElementCMove(t, &minusT, bNegative)
+}
+
+// geScalarMultBase computes h = a*B, where
+// a = a[0]+256*a[1]+...+256^31 a[31]
+// B is the Ed25519 base point (x,4/5) with x positive.
+//
+// Preconditions:
+// a[31] <= 127
+func geScalarMultBase(h *extendedGroupElement, a *[32]byte) {
+ var e [64]int8
+
+ for i, v := range a {
+ e[2*i] = int8(v & 15)
+ e[2*i+1] = int8((v >> 4) & 15)
+ }
+
+ // each e[i] is between 0 and 15 and e[63] is between 0 and 7.
+
+ carry := int8(0)
+ for i := 0; i < 63; i++ {
+ e[i] += carry
+ carry = (e[i] + 8) >> 4
+ e[i] -= carry << 4
+ }
+ e[63] += carry
+ // each e[i] is between -8 and 8.
+
+ h.Zero()
+ var t preComputedGroupElement
+ var r completedGroupElement
+ for i := int32(1); i < 64; i += 2 {
+ selectPoint(&t, i/2, int32(e[i]))
+ geMixedAdd(&r, h, &t)
+ r.ToExtended(h)
+ }
+
+ var s projectiveGroupElement
+
+ h.Double(&r)
+ r.ToProjective(&s)
+ s.Double(&r)
+ r.ToProjective(&s)
+ s.Double(&r)
+ r.ToProjective(&s)
+ s.Double(&r)
+ r.ToExtended(h)
+
+ for i := int32(0); i < 64; i += 2 {
+ selectPoint(&t, i/2, int32(e[i]))
+ geMixedAdd(&r, h, &t)
+ r.ToExtended(h)
+ }
+}
+
+// The scalars are GF(2^252 + 27742317777372353535851937790883648493).
+
+// Input:
+// a[0]+256*a[1]+...+256^31*a[31] = a
+// b[0]+256*b[1]+...+256^31*b[31] = b
+// c[0]+256*c[1]+...+256^31*c[31] = c
+//
+// Output:
+// s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
+// where l = 2^252 + 27742317777372353535851937790883648493.
+func scMulAdd(s, a, b, c *[32]byte) {
+ a0 := 2097151 & load3(a[:])
+ a1 := 2097151 & (load4(a[2:]) >> 5)
+ a2 := 2097151 & (load3(a[5:]) >> 2)
+ a3 := 2097151 & (load4(a[7:]) >> 7)
+ a4 := 2097151 & (load4(a[10:]) >> 4)
+ a5 := 2097151 & (load3(a[13:]) >> 1)
+ a6 := 2097151 & (load4(a[15:]) >> 6)
+ a7 := 2097151 & (load3(a[18:]) >> 3)
+ a8 := 2097151 & load3(a[21:])
+ a9 := 2097151 & (load4(a[23:]) >> 5)
+ a10 := 2097151 & (load3(a[26:]) >> 2)
+ a11 := (load4(a[28:]) >> 7)
+ b0 := 2097151 & load3(b[:])
+ b1 := 2097151 & (load4(b[2:]) >> 5)
+ b2 := 2097151 & (load3(b[5:]) >> 2)
+ b3 := 2097151 & (load4(b[7:]) >> 7)
+ b4 := 2097151 & (load4(b[10:]) >> 4)
+ b5 := 2097151 & (load3(b[13:]) >> 1)
+ b6 := 2097151 & (load4(b[15:]) >> 6)
+ b7 := 2097151 & (load3(b[18:]) >> 3)
+ b8 := 2097151 & load3(b[21:])
+ b9 := 2097151 & (load4(b[23:]) >> 5)
+ b10 := 2097151 & (load3(b[26:]) >> 2)
+ b11 := (load4(b[28:]) >> 7)
+ c0 := 2097151 & load3(c[:])
+ c1 := 2097151 & (load4(c[2:]) >> 5)
+ c2 := 2097151 & (load3(c[5:]) >> 2)
+ c3 := 2097151 & (load4(c[7:]) >> 7)
+ c4 := 2097151 & (load4(c[10:]) >> 4)
+ c5 := 2097151 & (load3(c[13:]) >> 1)
+ c6 := 2097151 & (load4(c[15:]) >> 6)
+ c7 := 2097151 & (load3(c[18:]) >> 3)
+ c8 := 2097151 & load3(c[21:])
+ c9 := 2097151 & (load4(c[23:]) >> 5)
+ c10 := 2097151 & (load3(c[26:]) >> 2)
+ c11 := (load4(c[28:]) >> 7)
+ var carry [23]int64
+
+ s0 := c0 + a0*b0
+ s1 := c1 + a0*b1 + a1*b0
+ s2 := c2 + a0*b2 + a1*b1 + a2*b0
+ s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0
+ s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0
+ s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0
+ s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0
+ s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0
+ s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0
+ s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0
+ s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0
+ s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0
+ s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1
+ s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2
+ s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3
+ s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4
+ s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5
+ s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6
+ s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7
+ s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8
+ s20 := a9*b11 + a10*b10 + a11*b9
+ s21 := a10*b11 + a11*b10
+ s22 := a11 * b11
+ s23 := int64(0)
+
+ carry[0] = (s0 + (1 << 20)) >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[2] = (s2 + (1 << 20)) >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[4] = (s4 + (1 << 20)) >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[6] = (s6 + (1 << 20)) >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[8] = (s8 + (1 << 20)) >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[10] = (s10 + (1 << 20)) >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+ carry[12] = (s12 + (1 << 20)) >> 21
+ s13 += carry[12]
+ s12 -= carry[12] << 21
+ carry[14] = (s14 + (1 << 20)) >> 21
+ s15 += carry[14]
+ s14 -= carry[14] << 21
+ carry[16] = (s16 + (1 << 20)) >> 21
+ s17 += carry[16]
+ s16 -= carry[16] << 21
+ carry[18] = (s18 + (1 << 20)) >> 21
+ s19 += carry[18]
+ s18 -= carry[18] << 21
+ carry[20] = (s20 + (1 << 20)) >> 21
+ s21 += carry[20]
+ s20 -= carry[20] << 21
+ carry[22] = (s22 + (1 << 20)) >> 21
+ s23 += carry[22]
+ s22 -= carry[22] << 21
+
+ carry[1] = (s1 + (1 << 20)) >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[3] = (s3 + (1 << 20)) >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[5] = (s5 + (1 << 20)) >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[7] = (s7 + (1 << 20)) >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[9] = (s9 + (1 << 20)) >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[11] = (s11 + (1 << 20)) >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+ carry[13] = (s13 + (1 << 20)) >> 21
+ s14 += carry[13]
+ s13 -= carry[13] << 21
+ carry[15] = (s15 + (1 << 20)) >> 21
+ s16 += carry[15]
+ s15 -= carry[15] << 21
+ carry[17] = (s17 + (1 << 20)) >> 21
+ s18 += carry[17]
+ s17 -= carry[17] << 21
+ carry[19] = (s19 + (1 << 20)) >> 21
+ s20 += carry[19]
+ s19 -= carry[19] << 21
+ carry[21] = (s21 + (1 << 20)) >> 21
+ s22 += carry[21]
+ s21 -= carry[21] << 21
+
+ s11 += s23 * 666643
+ s12 += s23 * 470296
+ s13 += s23 * 654183
+ s14 -= s23 * 997805
+ s15 += s23 * 136657
+ s16 -= s23 * 683901
+ s23 = 0
+
+ s10 += s22 * 666643
+ s11 += s22 * 470296
+ s12 += s22 * 654183
+ s13 -= s22 * 997805
+ s14 += s22 * 136657
+ s15 -= s22 * 683901
+ s22 = 0
+
+ s9 += s21 * 666643
+ s10 += s21 * 470296
+ s11 += s21 * 654183
+ s12 -= s21 * 997805
+ s13 += s21 * 136657
+ s14 -= s21 * 683901
+ s21 = 0
+
+ s8 += s20 * 666643
+ s9 += s20 * 470296
+ s10 += s20 * 654183
+ s11 -= s20 * 997805
+ s12 += s20 * 136657
+ s13 -= s20 * 683901
+ s20 = 0
+
+ s7 += s19 * 666643
+ s8 += s19 * 470296
+ s9 += s19 * 654183
+ s10 -= s19 * 997805
+ s11 += s19 * 136657
+ s12 -= s19 * 683901
+ s19 = 0
+
+ s6 += s18 * 666643
+ s7 += s18 * 470296
+ s8 += s18 * 654183
+ s9 -= s18 * 997805
+ s10 += s18 * 136657
+ s11 -= s18 * 683901
+ s18 = 0
+
+ carry[6] = (s6 + (1 << 20)) >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[8] = (s8 + (1 << 20)) >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[10] = (s10 + (1 << 20)) >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+ carry[12] = (s12 + (1 << 20)) >> 21
+ s13 += carry[12]
+ s12 -= carry[12] << 21
+ carry[14] = (s14 + (1 << 20)) >> 21
+ s15 += carry[14]
+ s14 -= carry[14] << 21
+ carry[16] = (s16 + (1 << 20)) >> 21
+ s17 += carry[16]
+ s16 -= carry[16] << 21
+
+ carry[7] = (s7 + (1 << 20)) >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[9] = (s9 + (1 << 20)) >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[11] = (s11 + (1 << 20)) >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+ carry[13] = (s13 + (1 << 20)) >> 21
+ s14 += carry[13]
+ s13 -= carry[13] << 21
+ carry[15] = (s15 + (1 << 20)) >> 21
+ s16 += carry[15]
+ s15 -= carry[15] << 21
+
+ s5 += s17 * 666643
+ s6 += s17 * 470296
+ s7 += s17 * 654183
+ s8 -= s17 * 997805
+ s9 += s17 * 136657
+ s10 -= s17 * 683901
+ s17 = 0
+
+ s4 += s16 * 666643
+ s5 += s16 * 470296
+ s6 += s16 * 654183
+ s7 -= s16 * 997805
+ s8 += s16 * 136657
+ s9 -= s16 * 683901
+ s16 = 0
+
+ s3 += s15 * 666643
+ s4 += s15 * 470296
+ s5 += s15 * 654183
+ s6 -= s15 * 997805
+ s7 += s15 * 136657
+ s8 -= s15 * 683901
+ s15 = 0
+
+ s2 += s14 * 666643
+ s3 += s14 * 470296
+ s4 += s14 * 654183
+ s5 -= s14 * 997805
+ s6 += s14 * 136657
+ s7 -= s14 * 683901
+ s14 = 0
+
+ s1 += s13 * 666643
+ s2 += s13 * 470296
+ s3 += s13 * 654183
+ s4 -= s13 * 997805
+ s5 += s13 * 136657
+ s6 -= s13 * 683901
+ s13 = 0
+
+ s0 += s12 * 666643
+ s1 += s12 * 470296
+ s2 += s12 * 654183
+ s3 -= s12 * 997805
+ s4 += s12 * 136657
+ s5 -= s12 * 683901
+ s12 = 0
+
+ carry[0] = (s0 + (1 << 20)) >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[2] = (s2 + (1 << 20)) >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[4] = (s4 + (1 << 20)) >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[6] = (s6 + (1 << 20)) >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[8] = (s8 + (1 << 20)) >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[10] = (s10 + (1 << 20)) >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+
+ carry[1] = (s1 + (1 << 20)) >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[3] = (s3 + (1 << 20)) >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[5] = (s5 + (1 << 20)) >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[7] = (s7 + (1 << 20)) >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[9] = (s9 + (1 << 20)) >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[11] = (s11 + (1 << 20)) >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+
+ s0 += s12 * 666643
+ s1 += s12 * 470296
+ s2 += s12 * 654183
+ s3 -= s12 * 997805
+ s4 += s12 * 136657
+ s5 -= s12 * 683901
+ s12 = 0
+
+ carry[0] = s0 >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[1] = s1 >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[2] = s2 >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[3] = s3 >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[4] = s4 >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[5] = s5 >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[6] = s6 >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[7] = s7 >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[8] = s8 >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[9] = s9 >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[10] = s10 >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+ carry[11] = s11 >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+
+ s0 += s12 * 666643
+ s1 += s12 * 470296
+ s2 += s12 * 654183
+ s3 -= s12 * 997805
+ s4 += s12 * 136657
+ s5 -= s12 * 683901
+ s12 = 0
+
+ carry[0] = s0 >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[1] = s1 >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[2] = s2 >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[3] = s3 >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[4] = s4 >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[5] = s5 >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[6] = s6 >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[7] = s7 >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[8] = s8 >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[9] = s9 >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[10] = s10 >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+
+ s[0] = byte(s0 >> 0)
+ s[1] = byte(s0 >> 8)
+ s[2] = byte((s0 >> 16) | (s1 << 5))
+ s[3] = byte(s1 >> 3)
+ s[4] = byte(s1 >> 11)
+ s[5] = byte((s1 >> 19) | (s2 << 2))
+ s[6] = byte(s2 >> 6)
+ s[7] = byte((s2 >> 14) | (s3 << 7))
+ s[8] = byte(s3 >> 1)
+ s[9] = byte(s3 >> 9)
+ s[10] = byte((s3 >> 17) | (s4 << 4))
+ s[11] = byte(s4 >> 4)
+ s[12] = byte(s4 >> 12)
+ s[13] = byte((s4 >> 20) | (s5 << 1))
+ s[14] = byte(s5 >> 7)
+ s[15] = byte((s5 >> 15) | (s6 << 6))
+ s[16] = byte(s6 >> 2)
+ s[17] = byte(s6 >> 10)
+ s[18] = byte((s6 >> 18) | (s7 << 3))
+ s[19] = byte(s7 >> 5)
+ s[20] = byte(s7 >> 13)
+ s[21] = byte(s8 >> 0)
+ s[22] = byte(s8 >> 8)
+ s[23] = byte((s8 >> 16) | (s9 << 5))
+ s[24] = byte(s9 >> 3)
+ s[25] = byte(s9 >> 11)
+ s[26] = byte((s9 >> 19) | (s10 << 2))
+ s[27] = byte(s10 >> 6)
+ s[28] = byte((s10 >> 14) | (s11 << 7))
+ s[29] = byte(s11 >> 1)
+ s[30] = byte(s11 >> 9)
+ s[31] = byte(s11 >> 17)
+}
+
+// Input:
+// s[0]+256*s[1]+...+256^63*s[63] = s
+//
+// Output:
+// s[0]+256*s[1]+...+256^31*s[31] = s mod l
+// where l = 2^252 + 27742317777372353535851937790883648493.
+func scReduce(out *[32]byte, s *[64]byte) {
+ s0 := 2097151 & load3(s[:])
+ s1 := 2097151 & (load4(s[2:]) >> 5)
+ s2 := 2097151 & (load3(s[5:]) >> 2)
+ s3 := 2097151 & (load4(s[7:]) >> 7)
+ s4 := 2097151 & (load4(s[10:]) >> 4)
+ s5 := 2097151 & (load3(s[13:]) >> 1)
+ s6 := 2097151 & (load4(s[15:]) >> 6)
+ s7 := 2097151 & (load3(s[18:]) >> 3)
+ s8 := 2097151 & load3(s[21:])
+ s9 := 2097151 & (load4(s[23:]) >> 5)
+ s10 := 2097151 & (load3(s[26:]) >> 2)
+ s11 := 2097151 & (load4(s[28:]) >> 7)
+ s12 := 2097151 & (load4(s[31:]) >> 4)
+ s13 := 2097151 & (load3(s[34:]) >> 1)
+ s14 := 2097151 & (load4(s[36:]) >> 6)
+ s15 := 2097151 & (load3(s[39:]) >> 3)
+ s16 := 2097151 & load3(s[42:])
+ s17 := 2097151 & (load4(s[44:]) >> 5)
+ s18 := 2097151 & (load3(s[47:]) >> 2)
+ s19 := 2097151 & (load4(s[49:]) >> 7)
+ s20 := 2097151 & (load4(s[52:]) >> 4)
+ s21 := 2097151 & (load3(s[55:]) >> 1)
+ s22 := 2097151 & (load4(s[57:]) >> 6)
+ s23 := (load4(s[60:]) >> 3)
+
+ s11 += s23 * 666643
+ s12 += s23 * 470296
+ s13 += s23 * 654183
+ s14 -= s23 * 997805
+ s15 += s23 * 136657
+ s16 -= s23 * 683901
+ s23 = 0
+
+ s10 += s22 * 666643
+ s11 += s22 * 470296
+ s12 += s22 * 654183
+ s13 -= s22 * 997805
+ s14 += s22 * 136657
+ s15 -= s22 * 683901
+ s22 = 0
+
+ s9 += s21 * 666643
+ s10 += s21 * 470296
+ s11 += s21 * 654183
+ s12 -= s21 * 997805
+ s13 += s21 * 136657
+ s14 -= s21 * 683901
+ s21 = 0
+
+ s8 += s20 * 666643
+ s9 += s20 * 470296
+ s10 += s20 * 654183
+ s11 -= s20 * 997805
+ s12 += s20 * 136657
+ s13 -= s20 * 683901
+ s20 = 0
+
+ s7 += s19 * 666643
+ s8 += s19 * 470296
+ s9 += s19 * 654183
+ s10 -= s19 * 997805
+ s11 += s19 * 136657
+ s12 -= s19 * 683901
+ s19 = 0
+
+ s6 += s18 * 666643
+ s7 += s18 * 470296
+ s8 += s18 * 654183
+ s9 -= s18 * 997805
+ s10 += s18 * 136657
+ s11 -= s18 * 683901
+ s18 = 0
+
+ var carry [17]int64
+
+ carry[6] = (s6 + (1 << 20)) >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[8] = (s8 + (1 << 20)) >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[10] = (s10 + (1 << 20)) >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+ carry[12] = (s12 + (1 << 20)) >> 21
+ s13 += carry[12]
+ s12 -= carry[12] << 21
+ carry[14] = (s14 + (1 << 20)) >> 21
+ s15 += carry[14]
+ s14 -= carry[14] << 21
+ carry[16] = (s16 + (1 << 20)) >> 21
+ s17 += carry[16]
+ s16 -= carry[16] << 21
+
+ carry[7] = (s7 + (1 << 20)) >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[9] = (s9 + (1 << 20)) >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[11] = (s11 + (1 << 20)) >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+ carry[13] = (s13 + (1 << 20)) >> 21
+ s14 += carry[13]
+ s13 -= carry[13] << 21
+ carry[15] = (s15 + (1 << 20)) >> 21
+ s16 += carry[15]
+ s15 -= carry[15] << 21
+
+ s5 += s17 * 666643
+ s6 += s17 * 470296
+ s7 += s17 * 654183
+ s8 -= s17 * 997805
+ s9 += s17 * 136657
+ s10 -= s17 * 683901
+ s17 = 0
+
+ s4 += s16 * 666643
+ s5 += s16 * 470296
+ s6 += s16 * 654183
+ s7 -= s16 * 997805
+ s8 += s16 * 136657
+ s9 -= s16 * 683901
+ s16 = 0
+
+ s3 += s15 * 666643
+ s4 += s15 * 470296
+ s5 += s15 * 654183
+ s6 -= s15 * 997805
+ s7 += s15 * 136657
+ s8 -= s15 * 683901
+ s15 = 0
+
+ s2 += s14 * 666643
+ s3 += s14 * 470296
+ s4 += s14 * 654183
+ s5 -= s14 * 997805
+ s6 += s14 * 136657
+ s7 -= s14 * 683901
+ s14 = 0
+
+ s1 += s13 * 666643
+ s2 += s13 * 470296
+ s3 += s13 * 654183
+ s4 -= s13 * 997805
+ s5 += s13 * 136657
+ s6 -= s13 * 683901
+ s13 = 0
+
+ s0 += s12 * 666643
+ s1 += s12 * 470296
+ s2 += s12 * 654183
+ s3 -= s12 * 997805
+ s4 += s12 * 136657
+ s5 -= s12 * 683901
+ s12 = 0
+
+ carry[0] = (s0 + (1 << 20)) >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[2] = (s2 + (1 << 20)) >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[4] = (s4 + (1 << 20)) >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[6] = (s6 + (1 << 20)) >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[8] = (s8 + (1 << 20)) >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[10] = (s10 + (1 << 20)) >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+
+ carry[1] = (s1 + (1 << 20)) >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[3] = (s3 + (1 << 20)) >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[5] = (s5 + (1 << 20)) >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[7] = (s7 + (1 << 20)) >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[9] = (s9 + (1 << 20)) >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[11] = (s11 + (1 << 20)) >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+
+ s0 += s12 * 666643
+ s1 += s12 * 470296
+ s2 += s12 * 654183
+ s3 -= s12 * 997805
+ s4 += s12 * 136657
+ s5 -= s12 * 683901
+ s12 = 0
+
+ carry[0] = s0 >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[1] = s1 >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[2] = s2 >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[3] = s3 >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[4] = s4 >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[5] = s5 >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[6] = s6 >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[7] = s7 >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[8] = s8 >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[9] = s9 >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[10] = s10 >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+ carry[11] = s11 >> 21
+ s12 += carry[11]
+ s11 -= carry[11] << 21
+
+ s0 += s12 * 666643
+ s1 += s12 * 470296
+ s2 += s12 * 654183
+ s3 -= s12 * 997805
+ s4 += s12 * 136657
+ s5 -= s12 * 683901
+ s12 = 0
+
+ carry[0] = s0 >> 21
+ s1 += carry[0]
+ s0 -= carry[0] << 21
+ carry[1] = s1 >> 21
+ s2 += carry[1]
+ s1 -= carry[1] << 21
+ carry[2] = s2 >> 21
+ s3 += carry[2]
+ s2 -= carry[2] << 21
+ carry[3] = s3 >> 21
+ s4 += carry[3]
+ s3 -= carry[3] << 21
+ carry[4] = s4 >> 21
+ s5 += carry[4]
+ s4 -= carry[4] << 21
+ carry[5] = s5 >> 21
+ s6 += carry[5]
+ s5 -= carry[5] << 21
+ carry[6] = s6 >> 21
+ s7 += carry[6]
+ s6 -= carry[6] << 21
+ carry[7] = s7 >> 21
+ s8 += carry[7]
+ s7 -= carry[7] << 21
+ carry[8] = s8 >> 21
+ s9 += carry[8]
+ s8 -= carry[8] << 21
+ carry[9] = s9 >> 21
+ s10 += carry[9]
+ s9 -= carry[9] << 21
+ carry[10] = s10 >> 21
+ s11 += carry[10]
+ s10 -= carry[10] << 21
+
+ out[0] = byte(s0 >> 0)
+ out[1] = byte(s0 >> 8)
+ out[2] = byte((s0 >> 16) | (s1 << 5))
+ out[3] = byte(s1 >> 3)
+ out[4] = byte(s1 >> 11)
+ out[5] = byte((s1 >> 19) | (s2 << 2))
+ out[6] = byte(s2 >> 6)
+ out[7] = byte((s2 >> 14) | (s3 << 7))
+ out[8] = byte(s3 >> 1)
+ out[9] = byte(s3 >> 9)
+ out[10] = byte((s3 >> 17) | (s4 << 4))
+ out[11] = byte(s4 >> 4)
+ out[12] = byte(s4 >> 12)
+ out[13] = byte((s4 >> 20) | (s5 << 1))
+ out[14] = byte(s5 >> 7)
+ out[15] = byte((s5 >> 15) | (s6 << 6))
+ out[16] = byte(s6 >> 2)
+ out[17] = byte(s6 >> 10)
+ out[18] = byte((s6 >> 18) | (s7 << 3))
+ out[19] = byte(s7 >> 5)
+ out[20] = byte(s7 >> 13)
+ out[21] = byte(s8 >> 0)
+ out[22] = byte(s8 >> 8)
+ out[23] = byte((s8 >> 16) | (s9 << 5))
+ out[24] = byte(s9 >> 3)
+ out[25] = byte(s9 >> 11)
+ out[26] = byte((s9 >> 19) | (s10 << 2))
+ out[27] = byte(s10 >> 6)
+ out[28] = byte((s10 >> 14) | (s11 << 7))
+ out[29] = byte(s11 >> 1)
+ out[30] = byte(s11 >> 9)
+ out[31] = byte(s11 >> 17)
+}
+
+// GenerateKey generates a public/private key pair using randomness from rand.
+func GenerateKey(rand io.Reader) (publicKey *[PublicKeySize]byte, privateKey *[PrivateKeySize]byte, err error) {
+ privateKey = new([64]byte)
+ publicKey = new([32]byte)
+ _, err = io.ReadFull(rand, privateKey[:32])
+ if err != nil {
+ return nil, nil, err
+ }
+
+ h := sha512.New()
+ h.Write(privateKey[:32])
+ digest := h.Sum(nil)
+
+ digest[0] &= 248
+ digest[31] &= 127
+ digest[31] |= 64
+
+ var A extendedGroupElement
+ var hBytes [32]byte
+ copy(hBytes[:], digest)
+ geScalarMultBase(&A, &hBytes)
+ A.ToBytes(publicKey)
+
+ copy(privateKey[32:], publicKey[:])
+ return
+}
+
+// Sign signs the message with privateKey and returns a signature.
+func Sign(privateKey *[PrivateKeySize]byte, message []byte) *[SignatureSize]byte {
+ h := sha512.New()
+ h.Write(privateKey[:32])
+
+ var digest1, messageDigest, hramDigest [64]byte
+ var expandedSecretKey [32]byte
+ h.Sum(digest1[:0])
+ copy(expandedSecretKey[:], digest1[:])
+ expandedSecretKey[0] &= 248
+ expandedSecretKey[31] &= 63
+ expandedSecretKey[31] |= 64
+
+ h.Reset()
+ h.Write(digest1[32:])
+ h.Write(message)
+ h.Sum(messageDigest[:0])
+
+ var messageDigestReduced [32]byte
+ scReduce(&messageDigestReduced, &messageDigest)
+ var R extendedGroupElement
+ geScalarMultBase(&R, &messageDigestReduced)
+
+ var encodedR [32]byte
+ R.ToBytes(&encodedR)
+
+ h.Reset()
+ h.Write(encodedR[:])
+ h.Write(privateKey[32:])
+ h.Write(message)
+ h.Sum(hramDigest[:0])
+ var hramDigestReduced [32]byte
+ scReduce(&hramDigestReduced, &hramDigest)
+
+ var s [32]byte
+ scMulAdd(&s, &hramDigestReduced, &expandedSecretKey, &messageDigestReduced)
+
+ signature := new([64]byte)
+ copy(signature[:], encodedR[:])
+ copy(signature[32:], s[:])
+ return signature
+}
+
+// Verify returns true iff sig is a valid signature of message by publicKey.
+func Verify(publicKey *[PublicKeySize]byte, message []byte, sig *[SignatureSize]byte) bool {
+ if sig[63]&224 != 0 {
+ return false
+ }
+
+ var A extendedGroupElement
+ if !A.FromBytes(publicKey) {
+ return false
+ }
+
+ h := sha512.New()
+ h.Write(sig[:32])
+ h.Write(publicKey[:])
+ h.Write(message)
+ var digest [64]byte
+ h.Sum(digest[:0])
+
+ var hReduced [32]byte
+ scReduce(&hReduced, &digest)
+
+ var R projectiveGroupElement
+ var b [32]byte
+ copy(b[:], sig[32:])
+ geDoubleScalarMultVartime(&R, &hReduced, &A, &b)
+
+ var checkR [32]byte
+ R.ToBytes(&checkR)
+ return bytes.Equal(sig[:32], checkR[:])
+}
diff --git a/ed25519_ref.go b/ed25519_ref.go
deleted file mode 100644
index ef4dcdf..0000000
--- a/ed25519_ref.go
+++ /dev/null
@@ -1,377 +0,0 @@
-// Copyright 2012 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-// Package ed25519 implements the Ed25519 signature algorithm. See
-// http://ed25519.cr.yp.to/.
-package ed25519
-
-import (
- "bytes"
- "crypto/sha512"
- "io"
- "math/big"
-)
-
-const (
- PublicKeySize = 32
- PrivateKeySize = 64
- SignatureSize = 64
-)
-
-var (
- // p25519 is 2**255-19, a prime number.
- p25519 *big.Int
- // n25521 is p-2 = 2**255-21.
- n25521 *big.Int
- // n2523 is (p-5)/8 = 2**252-3
- n2523 *big.Int
- // sqrtm1 is sqrt(-1) mod p
- sqrtm1 *big.Int
- // order is the order of the Twisted Edward's group: 2**252 + 27742317777372353535851937790883648493.
- order *big.Int
- // d is -121665/121666 mod 2**255-19, a parameter of the Twisted Edward's curve.
- d *big.Int
- // k is 2*d mod 2**255-19
- k *big.Int
- // (bX, bY) is the base point of the curve.
- bX, bY *big.Int
- bigOne *big.Int
-)
-
-func init() {
- p25519, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed", 16)
- n25521, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeb", 16)
- n2523, _ = new(big.Int).SetString("ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd", 16)
- sqrtm1, _ = new(big.Int).SetString("2b8324804fc1df0b2b4d00993dfbd7a72f431806ad2fe478c4ee1b274a0ea0b0", 16)
- d, _ = new(big.Int).SetString("52036cee2b6ffe738cc740797779e89800700a4d4141d8ab75eb4dca135978a3", 16)
- k, _ = new(big.Int).SetString("2406d9dc56dffce7198e80f2eef3d13000e0149a8283b156ebd69b9426b2f159", 16)
- bX, _ = new(big.Int).SetString("216936d3cd6e53fec0a4e231fdd6dc5c692cc7609525a7b2c9562d608f25d51a", 16)
- bY, _ = new(big.Int).SetString("6666666666666666666666666666666666666666666666666666666666666658", 16)
- order, _ = new(big.Int).SetString("7237005577332262213973186563042994240857116359379907606001950938285454250989", 10)
- bigOne = big.NewInt(1)
-}
-
-// invert sets out = a^{-1} mod 2**255-19 by raising to the power of p-2 (see
-// Fermat's Little Theorem).
-func invert(out, a *big.Int) {
- out.Exp(a, n25521, p25519)
-}
-
-// context is used to provide a pool of allocated big.Int's for use during
-// scalar multiplication.
-type context struct {
- t1, t2, t3, t4, t5, t6, t7, t8 *big.Int
-}
-
-func newContext() *context {
- return &context{
- t1: new(big.Int),
- t2: new(big.Int),
- t3: new(big.Int),
- t4: new(big.Int),
- t5: new(big.Int),
- t6: new(big.Int),
- t7: new(big.Int),
- t8: new(big.Int),
- }
-}
-
-// GenerateKey generates a public/private key pair using randomness from rand.
-func GenerateKey(rand io.Reader) (publicKey *[PublicKeySize]byte, privateKey *[PrivateKeySize]byte, err error) {
- privateKey = new([64]byte)
- publicKey = new([32]byte)
- _, err = io.ReadFull(rand, privateKey[:32])
- if err != nil {
- return nil, nil, err
- }
-
- h := sha512.New()
- h.Write(privateKey[:32])
- digest := h.Sum(nil)
-
- digest[0] &= 248
- digest[31] &= 127
- digest[31] |= 64
- a := scalarFrom32Bytes(digest[:32])
- Ax, Ay := newContext().scalarMult(a, bX, bY).toAffine()
- encodePoint(publicKey, Ax, Ay)
-
- copy(privateKey[32:], publicKey[:])
- return
-}
-
-// a fieldElement is a point on the Twisted Edward's curve represented in
-// extended coordinates. An affine point (x', y') is encoded as x = x'/z,
-// y = y'/z, t = x'y'/z.
-type fieldElement struct {
- x, y, z, t *big.Int
-}
-
-func newFieldElement() *fieldElement {
- return &fieldElement{
- x: new(big.Int),
- y: new(big.Int),
- z: new(big.Int),
- t: new(big.Int),
- }
-}
-
-// toAffine converts from extended coordinates to affine coordinates.
-func (fe *fieldElement) toAffine() (x, y *big.Int) {
- zInv := new(big.Int)
- invert(zInv, fe.z)
- x = new(big.Int).Mul(zInv, fe.x)
- x.Mod(x, p25519)
- y = new(big.Int).Mul(zInv, fe.y)
- y.Mod(y, p25519)
- return
-}
-
-// pointAdd adds two curve points, a and b, and puts a+b into out.
-// See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3
-func (c *context) pointAdd(out, a, b *fieldElement) {
- c.t1.Sub(a.y, a.x)
- if c.t1.Sign() < 0 {
- c.t1.Add(c.t1, p25519)
- }
- c.t2.Sub(b.y, b.x)
- if c.t2.Sign() < 0 {
- c.t2.Add(c.t2, p25519)
- }
- c.t1.Mul(c.t1, c.t2) // A
-
- c.t2.Add(a.y, a.x)
- c.t3.Add(b.y, b.x)
- c.t2.Mul(c.t2, c.t3) // B
-
- c.t3.Mul(a.t, b.t)
- c.t3.Mul(c.t3, k) // C
-
- c.t4.Mul(a.z, b.z)
- c.t4.Lsh(c.t4, 1) // D
-
- c.t5.Sub(c.t2, c.t1) // E
- if c.t5.Sign() < 0 {
- c.t5.Add(c.t5, p25519)
- }
-
- c.t6.Sub(c.t4, c.t3) // F
- if c.t6.Sign() < 0 {
- c.t6.Add(c.t6, p25519)
- }
-
- c.t7.Add(c.t4, c.t3) // G
- c.t8.Add(c.t1, c.t2) // H
-
- out.x.Mul(c.t5, c.t6)
- out.x.Mod(out.x, p25519)
- out.y.Mul(c.t7, c.t8)
- out.y.Mod(out.y, p25519)
- out.t.Mul(c.t5, c.t8)
- out.t.Mod(out.t, p25519)
- out.z.Mul(c.t6, c.t7)
- out.z.Mod(out.z, p25519)
-}
-
-// scalarMult calculates scalar*(px, py) and returns the result in extended
-// coordinates.
-func (c *context) scalarMult(scalar, px, py *big.Int) *fieldElement {
- scalarBytes := scalar.Bytes()
- n := newFieldElement()
- p := newFieldElement()
- p.x.Set(px)
- p.y.Set(py)
- p.z.SetInt64(1)
- p.t.Mul(p.x, p.y)
- n.x.SetInt64(0)
- n.y.SetInt64(1)
- n.z.SetInt64(1)
- n.t.SetInt64(0)
-
- for _, x := range scalarBytes {
- for bit := 0; bit < 8; bit++ {
- c.pointAdd(n, n, n)
- if x&0x80 != 0 {
- c.pointAdd(n, n, p)
- }
- x <<= 1
- }
- }
-
- return n
-}
-
-// encodePoint encodes a point on the curve into 32 bytes by encoding the y
-// coordinate in the first 255 bits, followed by the parity of the
-// x-coordinate.
-func encodePoint(out *[32]byte, px, py *big.Int) {
- b := py.Bytes()
- for i := range b {
- out[i] = b[len(b)-(1+i)]
- }
- if px.Bit(0) == 1 {
- out[31] |= 0x80
- }
-}
-
-// scalarFrom64Bytes interprets v as a 64-byte, little-endian number and
-// returns an element of the scalar field.
-func scalarFrom64Bytes(v []byte) *big.Int {
- // math.big takes bytes in big-endian form so we have to reverse the
- // bytes.
- var reversedBytes [64]byte
- for i := 0; i < 64; i++ {
- reversedBytes[i] = v[63-i]
- }
- r := new(big.Int).SetBytes(reversedBytes[:])
- r.Mod(r, order)
- return r
-}
-
-// scalarFrom32Bytes interprets v as a 32-byte, little-endian number and
-// returns an element of the scalar field.
-func scalarFrom32Bytes(v []byte) *big.Int {
- // math.big takes bytes in big-endian form so we have to reverse the
- // bytes.
- var reversedBytes [32]byte
- for i := 0; i < 32; i++ {
- reversedBytes[i] = v[31-i]
- }
- n := new(big.Int).SetBytes(reversedBytes[:])
- n.Mod(n, order)
- return n
-}
-
-// encodeScalar encodes s in to out as a 32-byte, little-endian number.
-func encodeScalar(out []byte, s *big.Int) {
- b := s.Bytes()
- for i := 0; i < len(b); i++ {
- out[i] = b[len(b)-(1+i)]
- }
-}
-
-// Sign signs the message with privateKey and returns a signature.
-func Sign(privateKey *[PrivateKeySize]byte, message []byte) *[SignatureSize]byte {
- h := sha512.New()
- h.Write(privateKey[:32])
- var digestBytes1, digestBytes2, digestBytes3 [64]byte
- expandedSecretKey := h.Sum(digestBytes1[:0])
-
- expandedSecretKey[0] &= 248
- expandedSecretKey[31] &= 127
- expandedSecretKey[31] |= 64
- a := scalarFrom32Bytes(expandedSecretKey[:32])
-
- h.Reset()
- h.Write(expandedSecretKey[32:])
- h.Write(message)
- messageDigest := h.Sum(digestBytes2[:0])
-
- r := scalarFrom64Bytes(messageDigest)
- Rx, Ry := newContext().scalarMult(r, bX, bY).toAffine()
- var encodedR [32]byte
- encodePoint(&encodedR, Rx, Ry)
-
- h.Reset()
- h.Write(encodedR[:])
- h.Write(privateKey[32:])
- h.Write(message)
- hramDigest := h.Sum(digestBytes3[:0])
- s := scalarFrom64Bytes(hramDigest)
- s.Mul(s, a)
- s.Add(s, r)
- s.Mod(s, order)
-
- signature := new([64]byte)
- copy(signature[:32], encodedR[:])
- encodeScalar(signature[32:], s)
- return signature
-}
-
-// decodePoint unpacks a curve point, as encoded by encodePoint, and returns
-// its affine coordinates.
-func decodePoint(in *[32]byte) (x, y *big.Int, ok bool) {
- var bigEndian [32]byte
- for i := 0; i < 32; i++ {
- bigEndian[i] = in[31-i]
- }
- bigEndian[0] &= 0x7f
- y = new(big.Int).SetBytes(bigEndian[:])
-
- u := new(big.Int).Mul(y, y)
- u.Mod(u, p25519)
- v := new(big.Int).Set(u)
- u.Sub(u, bigOne)
- if u.Sign() < 0 {
- u.Add(u, p25519)
- }
- v.Mul(v, d)
- v.Add(v, bigOne)
- v.Mod(v, p25519)
-
- v2 := new(big.Int).Mul(v, v)
- v3 := new(big.Int).Mul(v2, v)
- v3.Mod(v3, p25519)
- v4 := new(big.Int).Mul(v2, v2)
- v7 := v4.Mul(v4, v3)
- b := v7.Mul(v7, u)
- b.Mod(b, p25519)
- b.Exp(b, n2523, p25519)
- b.Mul(b, v3)
- b.Mul(b, u)
- b.Mod(b, p25519)
-
- check := v2.Mul(b, b)
- check.Mul(check, v)
- check.Mod(check, p25519)
- if check.Cmp(u) != 0 {
- b.Mul(b, sqrtm1)
- b.Mod(b, p25519)
- check.Mul(b, b)
- check.Mul(check, v)
- check.Mod(check, p25519)
- }
-
- if check.Cmp(u) != 0 {
- return nil, nil, false
- }
-
- if b.Bit(0) != uint(in[31]>>7) {
- b.Neg(b)
- b.Add(b, p25519)
- }
-
- return b, y, true
-}
-
-// Verify returns true iff sig is a valid signature of message by publicKey.
-func Verify(publicKey *[PublicKeySize]byte, message []byte, sig *[SignatureSize]byte) bool {
- h := sha512.New()
- h.Write(sig[:32])
- h.Write(publicKey[:])
- h.Write(message)
- digest := h.Sum(nil)
- H := scalarFrom64Bytes(digest)
- S := scalarFrom32Bytes(sig[32:])
-
- Ax, Ay, ok := decodePoint(publicKey)
- if !ok {
- return false
- }
-
- c := newContext()
- SB := c.scalarMult(S, bX, bY)
- HRAMA := c.scalarMult(H, Ax, Ay)
- HRAMA.x.Neg(HRAMA.x)
- HRAMA.x.Add(HRAMA.x, p25519)
- HRAMA.t.Neg(HRAMA.t)
- HRAMA.t.Add(HRAMA.t, p25519)
- out := newFieldElement()
- c.pointAdd(out, SB, HRAMA)
- Px, Py := out.toAffine()
-
- var rPrime [32]byte
- encodePoint(&rPrime, Px, Py)
- return bytes.Equal(sig[:32], rPrime[:])
-}
diff --git a/ed25519_test.go b/ed25519_test.go
index b9580f5..0dc2b13 100644
--- a/ed25519_test.go
+++ b/ed25519_test.go
@@ -25,7 +25,6 @@ func (zeroReader) Read(buf []byte) (int, error) {
}
func TestSignVerify(t *testing.T) {
- return
var zero zeroReader
public, private, _ := GenerateKey(zero)
@@ -94,7 +93,7 @@ func TestGolden(t *testing.T) {
sig2 := Sign(&priv, msg)
if !bytes.Equal(sig, sig2[:]) {
- t.Errorf("different signature result on line %d", lineNo)
+ t.Errorf("different signature result on line %d: %x vs %x", lineNo, sig, sig2)
}
var pubKey [PublicKeySize]byte
--
Alioth's /usr/local/bin/git-commit-notice on /srv/git.debian.org/git/pkg-privacy/packages/golang-ed25519-dev.git
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